unit 3 recap: trigonomic and polar functions

Key points

• Angles: positive (counterclockwise) and negative (clockwise), no limits on values, measured in standard position.
• Radian measure: ratio of arc length to radius, significant for understanding angle sizes.
• Trigonometric Functions:
  • Sine: vertical displacement (y-coordinate) over radius.
  • Cosine: horizontal displacement (x-coordinate) over radius.
  • Tangent: slope of the terminal side, or rise (y-coordinate) over run (x-coordinate).

Important Concepts

• Unit Circle: Central to understanding sine and cosine values; sine equals y-coordinate, cosine equals x-coordinate on the unit circle.
• Graphs of Trigonometric Functions:
  • Sine and Cosine Graphs: Patterns of these functions across one full cycle (period) and their characteristics (amplitude, midline).
  • Transformations: Use of variables to affect graph properties (amplitude, period length, vertical translation).
• Trigonometric Identities:
  • Pythagorean Identity: \(\sin^2\theta + \cos^2\theta = 1\)
  • Reciprocal, Quotient, and Pythagorean derived identities used for solving equations and simplifying expressions.
  • Double Angle Identities: Particularly for (\sin(2\theta)) and (\cos(2\theta)).
• Inverse Trigonometric Functions: Addressing the conversion of y-coordinates to angles within restricted domains for functions to remain proper inverses.
• Solving Trigonometric Equations: Technique varies between isolated trig functions and when inverses are involved; considerations for solution domains.
• Transforming and Utilizing Polar Coordinates:
  • Conversion between rectangular and polar coordinates uses specific trigonometric relationships.
  • Polar graphs (circles, lemniscates, rose curves) characterized by key features determined by their equations.

Strategies for Study and Practice

• Utilize unit circle memorization for quick identification of sine and cosine values.
• Practice transformations on sine and cosine graphs, focusing on effects of amplitude, period adjustments, and vertical shifts.
• Repeatedly work with trigonometric identities to familiarize oneself with their applications in different contexts.
• Engage in problem-solving involving inverse trig functions, ensuring understanding of their restricted domains.
• Convert between polar and rectangular coordinates to strengthen conceptual understanding and practical skills.
• Examine various types of polar graphs, noting the influence of equation parameters on graph characteristics.